ACTIVITY: Describing an Exponential Function

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1 6. Eponential Functions eponential function? What are the characteristics of an ACTIVITY: Describing an Eponential Function Work with a partner. The graph below shows estimates of the population of Earth from 000 B.C. through 00 A.D. at 00-ear intervals. a. Describe the pattern. b. Did Earth s population increase b the same amount or the same percent for each 00-ear period? Eplain. c. Assume the pattern continued. Estimate Earth s population in 000. d. Use the Internet to find Earth s population in 000. Did the pattern continue? If not, wh did the pattern change? Population of Earth Millions of people B.C. 000 B.C. 000 B.C. 000 B.C. 000 B.C A.D. 000 A.D. Year COMMON CORE Eponential Functions In this lesson, ou will identif, evaluate, and graph eponential functions. Learning Standards A.REI. A.REI. F.BF. F.IF.e F.LE.a F.LE. 000 B.C. Civilization begins to develop in Mesopotamia. 000 B.C. Stonehenge is built in England. 000 B.C. Middle Kingdom in Egpt 8 Chapter 6 Eponential Equations and Functions

2 Math Practice Calculate Accuratel How can ou check the accurac of our answers? ACTIVITY: Modeling an Eponential Function Work with a partner. Use the following eponential function to complete the table. Compare the results with the data in Activit. P = (.06) t/00 Year t Population from Activit P 000 b.c b.c b.c b.c b.c. 000 B.C. Augustus Caesar controls most of the Mediterranean world. (Use t 0 to approimate B.C.) 00 b.c b.c b.c b.c b.c. 00 b.c a.d A.D. Song Dnast has about one-fifth of Earth s population. 000 a.d a.d. 00. IN YOUR OWN WORDS What are the characteristics of an eponential function?. Sketch the graph of each eponential function. Does the function match the characteristics ou described in Question? Eplain. a. = b. = () c. = (.) Use what ou learned about eponential functions to complete Eercises and on page 89. Section 6. Eponential Functions 8

3 6. Lesson Lesson Tutorials Ke Vocabular eponential function, p. 86 A function of the form = ab, where a 0, b, and b > 0 is an eponential function. The eponential function = ab is a nonlinear function that changes b equal factors over equal intervals. EXAMPLE Identifing Functions Does each table represent a linear or an eponential function? Eplain. a b As increases b, increases b. The rate of change is constant. So, the function is linear As increases b, is multiplied b. So, the function is eponential. EXAMPLE Evaluating Eponential Functions Evaluate each function for the given value of. a. = () ; = b. = (0.) ; = = () Write the function. = (0.) = () Substitute for. = (0.) = () Evaluate the power. = () = 0 Multipl. = Eercises 6 Does the table represent a linear or an eponential function? Eplain Evaluate the function when =, 0, and.. = (9). =.() 86 Chapter 6 Eponential Equations and Functions

4 EXAMPLE Graphing an Eponential Function Graph =. Describe the domain and range. Stud Tip In Eample, ou can substitute an value for. So, the domain is all real numbers. Step : Make a table of values. 0 8 Step : Plot the ordered pairs. Step : Draw a smooth curve through the points. 8 6 From the graph, ou can see that the domain is all real numbers and the range is all positive real numbers. EXAMPLE Graphing a Vertical Translation Graph = +. Describe the domain and range. Compare the graph to the graph of =. Remember In Section., ou learned that the graph of f () + k is a vertical translation of the graph of f (). Step : Make a table of values. 0 Step : Plot the ordered pairs. Step : Draw a smooth curve through the points From the graph, ou can see that the domain is all real numbers and the range is all real numbers greater than. The graph of = + is a translation units up of the graph of =. Eercises and 9 Graph the function. Describe the domain and range. ). = ( ) ). Describe the domain and range. Compare the graph to the graph of = ( ).. = 6. = ( 8. Graph = ( Section 6. Eponential Functions 8

5 Stud Tip For an eponential function of the form = ab, the -values change b a factor of b as increases b. Also notice that a is the -intercept. To find the -intercept of the graph of = ab, substitute 0 for. = ab 0 = a() = a So, the -intercept is a = () EXAMPLE Real-Life Application The graph represents a bacteria population after das. Stud Tip For help with rational eponents, see Section 6.. a. Write an eponential function that represents the population. Use the graph to make a table of values The -intercept is and the -values increase b a factor of as increases b. So, the population can be modeled b = (). b. Find the population after hours and after das. Population Bacteria Population (0, ) 00 (, ) 00 (, 9) 00 (, 8) Da (, 68) Population after hours Population after das hours = da = () Write the function. = () = () / Substitute for. = () = () Evaluate the power. = (0) = 6 Multipl. = 0 There are 6 bacteria after hours and 0 bacteria after das. Eercises A bacteria population after das can be represented b an eponential function whose graph passes through (0, 00) and (, 00). a. Write a function that represents the population. b. Find the population after 6 das. Does this bacteria population grow faster than the bacteria population in Eample? Eplain. 88 Chapter 6 Eponential Equations and Functions

6 6. Eercises Help with Homework. VOCABULARY Describe how linear and eponential functions change over equal intervals.. OPEN-ENDED Sketch an increasing eponential function whose graph has a -intercept of.. WHICH ONE DOESN T BELONG? Which equation does not belong with the other three? Eplain our reasoning. = f () = () f () = ( ) = () 9+(-6)= +(-)= +(-9)= 9+(-)= Sketch the graph of the eponential function.. =. = () Does the table represent a linear or an eponential function? Eplain Evaluate the function for the given value of. 0. = ; =. f () = () ; =. = () ; =. f () = 0. ; =. f () = (6) ; =. = () ; = 6. ERROR ANALYSIS Describe and correct the error in evaluating the function.. CALCULATOR You graph an eponential function on a calculator. You zoom in repeatedl at % of the screen size. The function = 0. represents the percent (in decimal form) of the original screen displa that ou see, where is the number of times ou zoom in. You zoom in twice. What percent of the original screen do ou see? g() = 6(0.) ; = g( ) = 6(0.) = = 9 Section 6. Eponential Functions 89

7 Match the function with its graph. 8. f () = () 9. = (0.) 0. = (.) A. 6 B. 6 6 C. 6 Graph the function. Describe the domain and range.. = 9. f () =. f () = ( ). LOGIC Describe the graph of = a() when a is (a) positive and (b) negative. (c) How does the graph change as a changes?. NUMBER SENSE Consider the graph of f () = (b). How do the graphs differ when b > and 0 < b <? 6. COYOTES A population of cootes in a national park triples ever 0 ears. The function = () represents the population, where is the number of 0-ear periods. a. Graph the function. Describe the domain and range. b. Find and interpret the -intercept. c. How man cootes are in the national park after 0 ears? Graph the function. Describe the domain and range. Compare the graph to the graph of =.. = 8. = + 9. = 0. REASONING Graph the function f () =. Then graph g() =. a. Describe the domain and range of each function. b. Find the -intercept of the graph of each function. c. How are the -intercept, domain, and range affected b the translation?. REASONING When does an eponential function intersect the -ais? Give an eample to justif our answer. 90 Chapter 6 Eponential Equations and Functions

8 Given g () = 0., find the value of k so that the graph is g () + k..... REASONING Graph g() = +. Compare the graph to the graph of f () =. Write an eponential function represented b the graph or table (0, 0.) (, ) (, ) (, ). (, ) (, ) (, ) 6 8 (0, 8) ART GALLERY The graph represents the number of visitors to a new art galler after months. a. Write an eponential function that represents this situation. b. Approimate the number of visitors after months.. SALES A sales report shows that 00 gas grills were purchased from a chain of hardware stores last ear. The store epects grill sales to increase 6% each ear. About how man grills does the store epect to sell in ear 6? Use an equation to justif our answer. Visitors Art Galler (, 60) (0, 0) (, ) (, 90) Month. The graph of g is a translation units up and units right of the graph of f () =. Write an equation for g. Write the percent as a decimal. (Skills Review Handbook). %. %. 0% 6. MULTIPLE CHOICE Which of the following is equivalent to 00(0.9)? (Skills Review Handbook) A 0.9 B 9. C 9 D 90 Section 6. Eponential Functions 9